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Duality Theory: Biduality in Nonconvex Optimization

Article Outline

Keywords

SuperLagrangian Duality
Nonconvex Primal and Dual Problems
D.C. Programming and Hamiltonian
See also
References

Keywords Duality - Nonconvex optimization - d.c. programming - SuperLagrangian - Biduality - Clarke duality - Hamiltonian system

It is known that in convex optimization, the Lagrangian associated with a constrained problem is usually a saddle function, which leads to the classical saddle Lagrange duality (i. e. the monoduality ) theory. In nonconvex optimization, a so-called superLagrangian was introduced in [1], which leads to a nice biduality theory in convex Hamiltonian systems and in the so-called d.c. programming .

SuperLagrangian Duality

Definition 1 Let L(x, y ∗) be an arbitrary given real-valued function on 𝒳 × 𝒴∗.
A function L: 𝒳 × 𝒴∗ → R is said to be a supercritical function (or a ∂+-function ) on 𝒳 × 𝒴∗ if it is concave in each of its arguments.
A function

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